Piecewise Symmetric Tensors

Abstract

Every square matrix is uniquely the sum of a symmetric matrix and a skew-symmetric matrix. We extend this familiar fact to higher-order tensors: every cubic $k$-tensor is uniquely the sum of an $m$-piecewise symmetric tensor and a $(k-m)$-piecewise skew-symmetric tensor, for each choice of $m\leq k$. We study these tensor spaces via the combinatorics of word descents and compositions. Under Schur--Weyl duality, the two spaces naturally decompose into descent representations of the symmetric group. In fact, we prove a variant of Solomon's classical decomposition of its group algebra that explains this decomposition. Our motivation stems from signature tensors in stochastic analysis, algebraic geometry and data science: more specifically, we show that our tensor space decompositions determine the vanishing ideals of the signatures of piecewise linear paths with a fixed number of segments.